多层介质中多导体拐角结构电容参数的提取

本文给出一种计算多层介质中多导体拐角互连结构的准静态电容参数的快速算法─—维数缩减技术（DRT），并和有限差分法相结合，快速、准确地提取了多层介质中多导体拐角结构的准静态电容矩阵。由于维数缩减技术能充分利用集成电路结构分层的特点，从而大大减少了计算所需的时间和存储空间。文中给出的计算结果与Ansoft软件的计算结果吻合得较好，而计算所需的时间和存储空间大大少于Ansoft软件。     

区域分裂法在提取多层介质中多导体三维复杂互连结构电磁参数中的应用

本文首次将区域分裂法（ＤＤＭ）用于多层介质中多导体三维复杂互连结构的电磁参数的提取，可以快速、准确地提取复杂互连结构的静态电容矩阵。由于区域分裂法能肥大问题化为若干独立的小问题，不仅可以缩小计算规划，而且可以在该算法框架下灵活地组合各种三维互连结构，具有很强的灵活性，再充分利用集成电路结构分层的特点，对各个小问题采用最恰当的计算方法，从而可大大减少整体设计所需的时间和存储空间。文中给出的计算结果Ａ     

Fast computation of the two-dimensional generalised Hartleytransforms

The two-dimensional generalised Hartley transforms (2-D GDHTs) are various half-sample generalised DHTs, and are used for computing the 2-D DHT and 2-D convolutions. Fast computation of 2-D GDHTs is achieved by solving (n₁+(n₀₁/2))k₁+(n₂+(n₀₂ /2))k₂=(n+(½))k mod N, n₀₁, n₀₂ =1 or 0. The kernel indexes on the left-hand side and on the right-hand side belong to the 2-D GDHTs and the 1-D H₃, respectively. This equation categorises N×N-point input into N groups which are the inputs of a 1-D N-point H₃. By decomposing to 2-D GDHTs, an N×N-point DHT requires a 3N/2ⁱ 1-D N/2ⁱ-point H₃, i=1, ..., log₂N-2. Thus, it has not only the same number of multiplications as that of the discrete Radon transform (DRT) and linear congruence, but also has fewer additions than the DRT. The distinct H ₃ transforms are independent, and hence parallel computation is feasible. The mapping is very regular, and can be extended to an n-dimensional GDHT or GDFT easily     

Capacitance computations in a multilayered dielectric medium usingclosed-form spatial Green's functions

An efficient method to compute the 2-D and 3-D capacitance matrices of multiconductor interconnects in a multilayered dielectric medium is presented. The method is based on an integral equation approach and assumes the quasi-static condition. It is applicable to conductors of arbitrary polygonal shape embedded in a multilayered dielectric medium with possible ground planes on the top or bottom of the dielectric layers. The computation time required to evaluate the space-domain Green's function for the multilayered medium, which involves an infinite summation, has been greatly reduced by obtaining a closed-form expression, which is derived by approximating the Green's function using a finite number of images in the spectral domain. Then the corresponding space-domain Green's functions are obtained using the proper closed-form integrations. In both 2-D and 3-D cases, the unknown surface charge density is represented by pulse basis functions, and the delta testing function (point matching) is used to solve the integral equation. The elements of the resulting matrix are computed using the closed-form formulation, avoiding any numerical integration. The presented method is compared with other published results and showed good agreement. Finally, the equivalent microstrip crossover capacitance is computed to illustrate the use of a combination of 2-D and 3-D Green's functions     

A generalized algorithm for the capacitance extraction of 3D VLSIinterconnects

In this paper, a generalized algorithm based upon the nonoverlapping domain decomposition method (NDDM) is presented for the capacitance extraction of three-dimensional (3-D) VLSI interconnects. The subdomains with conductors are analyzed by the finite-difference method, while the subdomains with pure dielectric layers are analyzed with the eigenmode expansion technique. The central processing unit time and memory size used by the NDDM are unrelated to the thickness of pure dielectric layers. NDDM's computing time grows as O(n) if the number of domain iterations is bounded. Also, benchmarks show that it is approximately 15 times less than those used by Ansoft's Maxwell SpiceLink. In addition, only a two-dimensional mesh is needed to analyze 3-D structures. This greatly reduces the algorithm complexity and makes it easy and straightforward to interface with layout automation software     

一种改进的IC互连线3D电容提取快速层级算法

快速层级算法(FHM)是边界元法求解3D电容积分方程的一种加速方法,该方法基于分层近似对电势系数矩阵隐式表示,使求解的时间复杂度降低到O(n).改进算法对FHM做了两点改进:(1)给出了分层近似的理论依据,这种分层依据适用于所有导体结构而无需重复试验.(2)利用层级关系,直接计算面电荷,避免了迭代过程,加速了电荷求解.一系列典型3D互连线结构的测试显示:改进后的算法不仅提高了求解精度,而且计算时间也减少到改进前的1/3.     

多层介质中通孔结构准静态电容参数的提取

本文采用维数缩减技术(DRT)提取了多层介质中通孔结构的准静态电容参数。由于该方法充分利用了集成电路结构分层性的特点,从而可以很方便地处理任意的介质层数和结构参数,而仅需很少的计算时间和内存.文中的计算结果与Ansoft软件结果符合较好。     

An efficient solver for the three-dimensional capacitance of theinterconnects in high speed digital circuit by the multiresolutionmethod of moments

The computation of the equivalent capacitances for three-dimensional (3-D) interconnects features large memory usage and long computing time. In this paper, a matrix sparsification approach based on multiresolution representation is applied with the method of moments (MoM) to calculate 3-D capacitances of interconnects in a layered media. Instead of direct expansion of the charge distribution by the orthogonal wavelet basis functions, the large full matrix resulting from discretization of the integral equations is taken as a discrete image and sparsified by two-dimensional (2-D) multiresolution representations. The inverse of the obtained sparse matrix is efficiently implemented by Schultz's iterative approach. Several numerical examples are given and the results obtained show that the proposed method significantly sparsifies the matrix equation and the capacitance parameters computed by the matrix equation with high sparsity agree well with the results of other reports and those computed by an established capacitance extractor FASTCAP     

A unified RLC model for high-speed on-chip interconnects

In this paper, we propose a compact on-chip interconnect model for full-chip simulation. The model consists of two components, a quasi-three-dimensional (3-D) capacitance model and an effective loop inductance model. In the capacitance model, we propose a novel concept of effective width (W/sub eff/) for a 3-D wire, which is derived from an analytical two-dimensional (2-D) model combined with a new analytical "wall-to-wall" model. The effective width provides a physics-based approach to decompose any 3-D structure into a series of 2-D segments, resulting in an efficient and accurate capacitance extraction. In the inductance model, we use an effective loop inductance approach for an analytic and hierarchical model construction. In particular, we show empirically that high-frequency signals (above multi-GHz) propagating through random signal lines can be approximated by a quasi-TEM mode relationship, leading to a simple way to extract the high-frequency inductance from the capacitance of the wire. Finally, the capacitance and inductance models are combined into a unified frequency-dependent RLC model, describing successfully the wide-band characteristics of on-chip interconnects up to 100 GHz. Non-orthogonal wire architecture is also investigated and included in the proposed model.     

Fast and accurate quasi-three-dimensional capacitance determinationof multilayer VLSI interconnects

A new fast and accurate capacitance determination methodology for intricate multilayer VLSI interconnects is presented. Since a multilayer interconnect structure is too complicated to be directly tractable, it is simplified by investigating charge distributions within the system. The quasi-three-dimensional (3-D) capacitances of the structure are then determined by combining a set of solid-ground-based two-dimensional (2-D) capacitances and shielding effects that can be independently calculated from the simplified structure. The shielding effects due to the neighboring lines of a line can be analytically determined from the given layout dimensions. The solid-ground-based 2-D capacitances can also be quickly computed from the simplified structure. Thus, the proposed capacitance determination methodology is much more cost-efficient than conventional 3-D-based methods. It is shown that the calculated quasi-3-D capacitances have excellent agreement with 3-D field-solver-based results within 5% error     

Theory of ballistic nanotransistors

Numerical simulations are used to guide the development of a simple analytical theory for ballistic field-effect transistors. When two-dimensional (2-D) electrostatic effects are small (and when the insulator capacitance is much less than the semiconductor (quantum) capacitance), the model reduces to Natori's theory of the ballistic MOSFET. The model also treats 2-D electrostatics and the quantum capacitance limit where the semiconductor quantum capacitance is much less than the insulator capacitance. This new model provides insights into the performance of MOSFETs near the scaling limit and a unified framework for assessing and comparing a variety of novel transistors.     

Capacitance extraction of integrated-circuit interconnects by matrix decomposition based on MEI concept

Recently, fast computational methods based on directly thinning the matrix of Method of Moment (MoM), have been introduced. In this paper, we propose a simply technique, based on the concept of Measured Equation of Invariance (MEI), to thin the MoM matrix numerically. This technique is referred to as Matrix Decomposition by MEI (MDMEI). A little effort is required to add the MDMEI to any of the variety of MoM programs for electromagnetic problems, such as the electrostatic problems, wire antennas, and two-dimensional (2-D) conducting object scattering. However, in this paper we only demonstrate how MDMEI is used in the capacitance extraction in integrated-circuit interconnects. The approach is verified by 2-D and 3-D examples with computing errors within 2–4%. For the present achievement, the sparsity rates of the resultant matrices in MDMEI depend on the problem to be solved. Further research is required to keep the bandwidths unchanged with object size increasing so that the sparsity rates of the MDMEI matrices are expected to improve.     

Investigation of interconnect capacitance characterization usingcharge-based capacitance measurement (CBCM) technique andthree-dimensional simulation

This paper examines the recently introduced charge-based capacitance measurement (CBCM) technique through use of a three-dimensional (3-D) interconnect simulator. This method can be used in conjunction with simulation at early process development stages to provide designers with accurate parasitic interconnect capacitances. Metal to substrate, interwire, and interlayer capacitances are each discussed and overall close agreement is found between CBCM and 3-D simulation. Full process interconnect characterization is one possible application of this new compact, high-resolution test structure     

ICCAP-a linear time sparsification and reordering algorithm for 3-D BEM capacitance extraction

This paper presents an efficient, simple, hierarchical, and sparse three-dimensional capacitance extraction algorithm, i.e., ICCAP. Most previous capacitance extraction algorithms, such as FastCap and HiCap, introduce intermediate variables to facilitate the hierarchical potential calculation, but still preserve the basic panels as basis. In this paper, we discover that those intermediate variables are a fundamentally much better basis than leaf panels. As a result, we are able to explicitly construct the sparse potential coefficient matrix and solve it with linear memory and linear run time in comparison with the most recent hierarchical O(nlogn) approach in PHiCap. Furthermore, the explicit sparse formulation of a potential matrix not only enables the usage of preconditioned Krylov subspace iterative methods, but also the reordering technique. A new reordering technique, i.e., level-oriented reordering (LOR), is proposed to further reduce over 20% of memory consumption and run time compared with no reordering techniques applied. In fact, LOR is even better than the state-of-the-art minimum degree reordering and more efficient. Without complicated orthonormalization matrix computation, ICCAP is very simple, efficient, and accurate. Experimental results demonstrate the superior run time and memory consumption over previous approaches while achieving similar accuracy.     

Fast modeling of microspherically focused log in a horizontallylayered formation

A microspherically focused log is a focused microresistivity device used for evaluation of the electrical property of the subsurface rock formation for oil and gas exploration. The electrodes are mounted on a flexible rubber pad that is applied against the borehole wall to be in close proximity to the formation. Unlike other nonpad-focused electrode tools that are centered in the borehole such as Dual-Laterolog, the modeling of the microspherically focused log is much more complex and requires three-dimensional (3-D) code in general because the electric field is no longer axially symmetric. However, in a horizontally layered medium, the axial symmetry of the earth formation makes the problem two-and-a-half-dimensional (2.5-D), which is much simpler. In this paper, the authors apply the semi-analytic method to tackle this 2.5-D problem and establish a fast-modeling algorithm. First, they expand the Green's function as a Fourier series in order to transform the 2.5-D problem into a sequence of the axially symmetric (2-D) problems. The semi-analytic method is used to obtain the expression of the Green's function. Applying the focusing conditions of the tool and the solution of the Green's function, we then establish the boundary integral equation with respect to the current density, currents, and potentials on the electrode surfaces, and study its numerical solutions. Then, they compare results computed from the semi-analytical method with those from the finite element method (FEM) to verify its accuracy. Finally, they study the response characteristics of the tool in several different environments by the semi-analytical method     

一种快速的三维VLSI互连电容提取方法:虚拟多介质方法

本文提出一种基于直接边界元方法的虚拟多介质(Quasi-Multiple Medium,QMM)加速方法,并将它应用于三维VLSI多介质互连电容的计算中.QMM方法将三维互连电容器中的单层介质看成由多个虚拟介质组成,从而大大减少了系数矩阵中的非零元数目,最终使计算时间和存储空间显著减少.通过比较QMM算法与非QMM算法,以及商业软件Raphael对实际三维互连结构的计算,结果表明QMM算法在保持计算准确性的同时,可使电容提取的效率得到显著提高.     

Markerless real-time 3-D target region tracking by motion backprojection from projection images

Accurate and fast localization of a predefined target region inside the patient is an important component of many image-guided therapy procedures. This problem is commonly solved by registration of intraoperative 2-D projection images to 3-D preoperative images. If the patient is not fixed during the intervention, the 2-D image acquisition is repeated several times during the procedure, and the registration problem can be cast instead as a 3-D tracking problem. To solve the 3-D problem, we propose in this paper to apply 2-D region tracking to first recover the components of the transformation that are in-plane to the projections. The 2-D motion estimates of all projections are backprojected into 3-D space, where they are then combined into a consistent estimate of the 3-D motion. We compare this method to intensity-based 2-D to 3-D registration and a combination of 2-D motion backprojection followed by a 2-D to 3-D registration stage. Using clinical data with a fiducial marker-based gold-standard transformation, we show that our method is capable of accurately tracking vertebral targets in 3-D from 2-D motion measured in X-ray projection images. Using a standard tracking algorithm (hyperplane tracking), tracking is achieved at video frame rates but fails relatively often (32% of all frames tracked with target registration error (TRE) better than 1.2 mm, 82% of all frames tracked with TRE better than 2.4 mm). With intensity-based 2-D to 2-D image registration using normalized mutual information (NMI) and pattern intensity (PI), accuracy and robustness are substantially improved. NMI tracked 82% of all frames in our data with TRE better than 1.2 mm and 96% of all frames with TRE better than 2.4 mm. This comes at the cost of a reduced frame rate, 1.7 s average processing time per frame and projection device. Results using PI were slightly more accurate, but required on average 5.4 s time per frame. These results are still substantially faster than 2-D to 3-D registration. We conclude that motion backprojection from 2-D motion tracking is an accurate and efficient method for tracking 3-D target motion, but tracking 2-D motion accurately and robustly remains a challenge.     

Capacitance extraction for electrostatic multiconductor problems byon-surface MEI

In this paper, on-surface measured equation of invariance (OSMEI) method is implemented for capacitance extraction of electrostatic multiconductor interconnect problems. OSMEI uses the same mesh as that in method of moments (MoM), but generates highly sparse matrices rather than a full matrix. In comparison with “standard” MEI which contains a few finite difference (FD) or finite element (FE) mesh layers, the number of unknowns and the computation memory can be saved. For each OSMEI equation in the multiconductor interconnects, a given node on a given conductor is forced into coupling with the few adjacent nodes on conductor itself and the few sampled nodes on other conductors. Thus, the system sparse matrices can be generated. The convergent behavior of the capacitance with the number of the nodes in the OSMEI equations has been widely investigated, Numerical examples of the capacitance extraction for two-dimensional (2-D) and three-dimensional (3-D) multiconductor interconnects show that the computing errors are within 24%. The OSMEI method may become a powerful technique for the more complex interconnect problems     

The effect of high-K gate dielectrics on deep submicrometer CMOSdevice and circuit performance

The potential impact of high permittivity gate dielectrics on device short channel and circuit performance is studied over a wide range of dielectric permittivities (K_gate) using two-dimensional (2-D) device and Monte Carlo simulations. The gate-to-channel capacitance and parasitic fringe capacitances are extracted using a highly accurate three-dimensional (3-D) capacitance extractor. It is observed that there is a decrease in parasitic outer fringe capacitance and gate-to-channel capacitance in addition to an increase in internal fringe capacitance, when the conventional silicon dioxide is replaced by a high-K gate dielectric. The lower parasitic outer fringe capacitance is beneficial for the circuit performance, while the increase in internal fringe capacitance and the decrease in the gate-to-channel capacitance will degrade the short channel performance contributing to higher DIBL, drain leakage, and lower noise margin. It is shown that using low-K gate sidewalls with high-K gate insulators can decrease the fringing-induced barrier lowering. Also, from the circuit point of view, for the 70-nm technology generation, the presence of an optimum K_gate for different target subthreshold leakage currents has been identified     

Consistency conditions and linear reconstruction methods in diffraction tomography

Because an image can be reconstructed from knowledge of its Radon transform (RT), the task of reconstructing an image is tantamount to that of estimating its RT. Based upon the Fourier diffraction projection (FDP) theorem, from the statistical perspective of unbiased reduction of image variance, we previously proposed an infinite family of estimation methods for obtaining the RT from the scattered data in diffraction tomography (DT). In this work, using the FDP theorem, we define the diffraction Radon transform (DRT), which can be treated as the data function in DT. Subsequently, using strategies similar to those that analyze the consistency conditions on the exponential Radon transform in two-dimensional (2-D) single-photon emission computed tomography with uniform attenuation, we studied the consistency condition on the DRT and we show that there is a hierarchy of estimation methods that actually project the noisy data function onto its consistency space in different ways. In terms of a weighted inner product of the consistency and inconsistency parts of a noisy data function, we further demonstrate that a subset of the family of estimation methods can be interpreted as orthogonal projections onto the consistency space of the DRT. In particular, the statistically suboptimal estimation method in the family corresponds to an orthogonal projection associated with an ordinary inner product of the consistency and inconsistency parts of a noisy data function.